5-1. Exercises

1. BASIC

Ex 1. A continuous random variable $X$ has a uniform distribution on the interval $[5,12]$ . Sketch the graph of its density function.

Ex 2. A continuous random variable $X$ has a uniform distribution on the interval $[−3,3]$ . Sketch the graph of its density function.

Ex 3. A continuous random variable $X$ has a normal distribution with mean 100 and standard deviation 10. Sketch a qualitatively accurate graph of its density function.

Ex 4. A continuous random variable $X$ has a normal distribution with mean 73 and standard deviation 2.5. Sketch a qualitatively accurate graph of its density function.

Ex 5. A continuous random variable $X$ has a normal distribution with mean 73. The probability that $X$ takes a value greater than 80 is 0.212. Use this information and the symmetry of the density function to find the probability that $X$ takes a value less than 66. Sketch the density curve with relevant regions shaded to illustrate the computation.

Ex 6. A continuous random variable $X$ has a normal distribution with mean 169. The probability that $X$ takes a value greater than 180 is 0.17. Use this information and the symmetry of the density function to find the probability that $X$ takes a value less than 158. Sketch the density curve with relevant regions shaded to illustrate the computation.

Ex 7. A continuous random variable $X$ has a normal distribution with mean 50.5. The probability that $X$ takes a value less than 54 is 0.76. Use this information and the symmetry of the density function to find the probability that $X$ takes a value greater than 47. Sketch the density curve with relevant regions shaded to illustrate the computation.

Ex 8. A continuous random variable $X$ has a normal distribution with mean 12.25. The probability that $X$ takes a value less than 13 is 0.82. Use this information and the symmetry of the density function to find the probability that $X$ takes a value greater than 11.50. Sketch the density curve with relevant regions shaded to illustrate the computation.

Ex 9. The figure provided shows the density curves of three normally distributed random variables $X_A$, $X_B$, and $X_C$. Their standard deviations (in no particular order) are 15, 7, and 20. Use the figure to identify the values of the means $μ_A, μ_B,$ and $μ_C$ and standard deviations $σ_A, σ_B,$ and $σ_C$ of the three random variables.

Ex 10. The figure provided shows the density curves of three normally distributed random variables $X_A$, $X_B$, and $X_C$. Their standard deviations (in no particular order) are 20, 5, and 10. Use the figure to identify the values of the means $μ_A, μ_B,$ and $μ_C$ and standard deviations $σ_A, σ_B,$ and $σ_C$ of the three random variables.

2. APPLICATIONS

Ex 11. Dogberry's alarm clock is battery operated. The battery could fail with equal probability at any time of the day or night. Every day Dogberry sets his alarm for 6:30 a.m. and goes to bed at 10:00 p.m. Find the probability that when the clock battery finally dies, it will do so at the most inconvenient time, between 10:00 p.m. and 6:30 a.m.

Ex 12. Buses running a bus line near Desdemona's house run every 15 minutes. Without paying attention to the schedule she walks to the nearest stop to take the bus to town. Find the probability that she waits more than 10 minutes.

Ex 13. The amount $X$ of orange juice in a randomly selected half-gallon container varies according to a normal distribution with mean 64 ounces and standard deviation 0.25 ounce.

1. Sketch the graph of the density function for $X$.

2. What proportion of all containers contain less than a half gallon (64 ounces)? Explain.

3. What is the median amount of orange juice in such containers? Explain.

Ex 14. The weight $X$ of grass seed in bags marked 50 lb varies according to a normal distribution with mean 50 lb and standard deviation 1 ounce (0.0625 lb).

1. Sketch the graph of the density function for $X$.

2. What proportion of all bags weigh less than 50 pounds? Explain.

3. What is the median weight of such bags? Explain.